Compressed recurrent neural network models

ABSTRACT

Methods, systems, and apparatus, including computer programs encoded on computer storage media, for implementing long-short term memory layers with compressed gating functions. One of the systems includes a first long short-term memory (LSTM) layer, wherein the first LSTM layer is configured to, for each of the plurality of time steps, generate a new layer state and a new layer output by applying a plurality of gates to a current layer input, a current layer state, and a current layer output, each of the plurality of gates being configured to, for each of the plurality of time steps, generate a respective intermediate gate output vector by multiplying a gate input vector and a gate parameter matrix. The gate parameter matrix for at least one of the plurality of gates is a structured matrix or is defined by a compressed parameter matrix and a projection matrix.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims the benefit of the filing date of U.S.Provisional Application No. 62/172,018, filed on Jun. 5, 2015, and U.S.Provisional Application No. 62/301,734, filed on Mar. 1, 2016. Thecontents of U.S. Application Nos. 62/172,018 and 62/301,734 areincorporated herein by reference in their entirety.

BACKGROUND

This specification relates to neural network architectures andcompressing neural networks.

Neural networks are machine learning models that employ one or morelayers of nonlinear units to predict an output for a received input.Some neural networks include one or more hidden layers in addition to anoutput layer. The output of each hidden layer is used as input to thenext layer in the network, i.e., the next hidden layer or the outputlayer. Each layer of the network generates an output from a receivedinput in accordance with current values of a respective set ofparameters. Some neural networks, e.g., those that are designed for timeseries problems or sequence-to-sequence learning (recurrent neuralnetworks (RNN)), incorporate recurrent loops which permit memory, in theform of a hidden state variable, to persist within a layer between datainputs. A variation of RNNs, long short-term memory (LSTM) neuralnetworks include multiple gates within each layer to control thepersistence of data between data inputs. Some neural networks, e.g.,those that are designed for time series problems or sequence-to-sequencelearning, incorporate recurrent loops which permit memory, in the formof a hidden state variable, to persist within a layer between datainputs.

SUMMARY

This specification describes technologies that relate to recurrentneural network architectures. In general, a recurrent neural networkincludes at least one Long Short-Term Memory (LSTM) layer that iscompressed. The LSTM layer has at least one gate that has a compressedparameter matrix. The LSTM layer may be compressed by either replacingone or more of the gate parameter matrices in the LSTM layer with aToeplitz-like structured matrix or by re-defining the gate parametermatrix with a compressed parameter matrix and a projection matrix.Optionally, one LSTM layer may be compressed by replacing one of thegate parameter matrices in the LSTM layer with a Toeplitz-likestructured matrix and another LSTM layer may be compressed by replacingone of the gate parameter matrices in the other LSTM layer byre-defining the gate parameter matrix with a compressed parameter matrixand a projection matrix.

For a system of one or more computers to be configured to performparticular operations or actions means that the system has installed onit software, firmware, hardware, or a combination of them that inoperation cause the system to perform the operations or actions. For oneor more computer programs to be configured to perform particularoperations or actions means that the one or more programs includeinstructions that, when executed by data processing apparatus, cause theapparatus to perform the operations or actions.

The subject matter described in this specification can be implemented inparticular embodiments so as to realize one or more of the followingadvantages. The performance of a recurrent neural network can beimproved by including a compressed LSTM layer in the recurrent neuralnetwork. In particular, by including the compressed LSTM layer in therecurrent neural network, the recurrent neural network is configured tobe able to process data more efficiently and use less data storage. Arecurrent neural network having a compressed LSTM layer can beeffectively trained to achieve word error rates comparable to full size,e.g., uncompressed, recurrent neural networks.

The details of one or more embodiments of the subject matter of thisspecification are set forth in the accompanying drawings and thedescription below. Other features, aspects, and advantages of thesubject matter will become apparent from the description, the drawings,and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows an example neural network system.

FIG. 2A and 2B show example structured matrices.

FIG. 3 is a flow diagram of an example process for processing a currentlayer input to generate a next layer output.

FIG. 4 is a flow diagram of an example process for applying a gate to agate input vector to generate a gate output vector.

FIG. 5 is a flow diagram of an example process for training a recurrentneural network that includes a saturating LSTM layer.

Like reference numbers and designations in the various drawings indicatelike elements.

DETAILED DESCRIPTION

FIG. 1 shows an example neural network system 100. The neural networksystem 100 is an example of a system implemented as computer programs onone or more computers in one or more locations, in which the systems,components, and techniques described below are implemented.

The neural network system 100 is a machine learning system that receivesa respective neural network input at each of multiple time steps andgenerates a respective neural network output at each of the time steps.That is, at each of the multiple time steps, the neural network system100 receives a neural network input and processes the neural networkinput to generate a neural network output. For example, at a given timestep, the neural network system 100 can receive a neural network input102 and generate a neural network output 132.

The neural network system 100 can store the generated neural networkoutputs in an output data repository or provide the neural networkoutputs for use for some other immediate purpose.

The neural network system 100 can be configured to receive any kind ofdigital data input and to generate any kind of score or classificationoutput based on the input.

For example, if the inputs to the neural network system 100 are imagesor features that have been extracted from images, the output generatedby the neural network system 100 for a given image may be scores foreach of a set of object categories, with each score representing anestimated likelihood that the image contains an image of an objectbelonging to the category.

As another example, if the inputs to the neural network system 100 areInternet resources (e.g., web pages), documents, or portions ofdocuments or features extracted from Internet resources, documents, orportions of documents, the output generated by the neural network system100 for a given Internet resource, document, or portion of a documentmay be a score for each of a set of topics, with each score representingan estimated likelihood that the Internet resource, document, ordocument portion is about the topic.

As another example, if the inputs to the neural network system 100 arefeatures of a personalized recommendation for a user, e.g., featurescharacterizing the context for the recommendation, e.g., featurescharacterizing previous actions taken by the user, the output generatedby the neural network system 100 may be a score for each of a set ofcontent items, with each score representing an estimated likelihood thatthe user will respond favorably to being recommended the content item.In some of these examples, the neural network system 100 is part of areinforcement learning system that provides content recommendations tousers.

As another example, if the input to the neural network system 100 istext in one language, the output generated by the neural network system100 may be a score for each of a set of pieces of text in anotherlanguage, with each score representing an estimated likelihood that thepiece of text in the other language is a proper translation of the inputtext into the other language.

As another example, if the input to the neural network system 100 isfeatures of a spoken utterance, the output generated by the neuralnetwork system 100 may be a score for each of a set of pieces of text,each score representing an estimated likelihood that the piece of textis the correct transcription for the utterance.

As another example, if the inputs to the neural network system 100 areimages, the output generated by the neural network system 100 may be ascore for each of a set of pieces of text, each score representing anestimated likelihood that the piece of text is text that is present inthe input image.

In particular, the neural network system 100 includes a recurrent neuralnetwork 110 which, in turn, includes a compressed Long Short-Term Memory(LSTM) layer 120. The recurrent neural network 110 is configured to, ateach of the time steps, receive the neural network input and to processthe neural network input to generate the neural network output at thetime step.

In addition to the compressed LSTM layer 120, the recurrent neuralnetwork 110 may include one or more other components, e.g., othercompressed LSTM layers, conventional LSTM layers, other recurrent neuralnetwork layers, other non-recurrent neural network layers, and so on.

For example, the recurrent neural network 100 may be a deep LSTM networkthat includes an input layer, multiple LSTM layers including thecompressed LSTM layer 120 arranged in an ordered stack one on top of oneanother, and an output layer that, at each time step, receives the layeroutput from the highest LSTM layer in the stack and, optionally, otherLSTM layers in the stack, and processes the layer output to generate theneural network output 132 at the time step.

The compressed LSTM layer 120 is configured to, at each of the timesteps, receive a current layer input 122 and to process the currentlayer input 122, a current layer state, and a current layer output togenerate a new layer output 126 and to update the current layer state togenerate a new layer state 124.

Depending on the configuration of the recurrent neural network 110, thecurrent layer input 122 may be the neural network input 102 or an outputgenerated by a different component of the recurrent neural network 110.

Additionally, for each time step after the first step, the current layerstate is the new layer state generated at the preceding time step andthe current layer output is the new layer output from the preceding timestep. For the first time step, the current layer state may be apredetermined initial layer state and the current layer output may be apredetermined initial layer output.

Depending on the configuration of the recurrent neural network 110, thenew layer output 126 may be provided as input to another LSTM layer inthe recurrent neural network 110, as input to a different type of neuralnetwork component, e.g., to an output layer or a different type ofneural network layer, or may be provided as the neural network output132 of the recurrent neural network 110.

In particular, the compressed LSTM layer 120 applies multiple gates tothe current layer input 122, the current layer state, and the currentlayer output to generate the new layer output 126 and to update thecurrent layer state to generate the new layer state 124, with at leastone of the gates including a compressed weight matrix. For example, atleast one layer of the layer stack can be compressed by replacing one ormore of the gate parameter matrices in the layer with a structuredmatrix (“structured matrix compression”) or by re-defining the gateparameter matrices with a compressed parameter matrix and a projectionmatrix (“projection compression”). The gates can include, but are notlimited to, e.g., an input gate, a forget gate, a cell state gate, or anoutput gate. In addition, each gate can include both inter-layer andrecurrent parameter matrices.

A structured matrix is an m×n matrix that can be described in fewer thanmn parameters. FIG. 2A illustrates examples of structured matrices. Forexample, structured matrices include the following general classes:Toeplitz matrices 200, Vandermonde matrices 202, and Cauchy matrices204. In particular, a Toeplitz matrix 200 is one in which the parametersare tied along diagonals. That is, Toeplitz matrices 200 have constantvalues along each of their diagonals. When the same property holds foranti-diagonals, the matrices 200 are called Hankel matrices. AVandermonde matrix 202 is one in which matrix entries of the thirdthrough n-th columns are defined by taking elementwise powers of entriesin the second column of the matrix. Similarly, a Cauchy matrix 204 isone that can be completely defined by two vectors (U and V). Eachelement a_(ij) of the Cauchy matrix 204 is defined by

$\frac{1}{( {u_{i} - v_{j}} )}.$

The use of such structured matrices to represent gate matrices incompressed LSTM layers can reduce the memory requirements for LSTMnetworks because they can be fully described by fewer than mnparameters. In addition, structured matrices can accelerate the trainingand processing of LSTM networks because they allow matrix-vector productand gradient computations to be performed faster.

The above general classes of structured matrices may be modified into astructured-like matrix for use in a compressed LSTM layer 120. Forexample, Toeplitz-like matrices are generalizations of the Toeplitzstructure that include products and inverses of Toeplitz matrices, andtheir linear combinations. Toeplitz-like matrices can be parameterizedas shown in FIG. 2B, as a sum over products of r circulant andskew-circulant matrices.

Referring again to FIG. 1, for convenience, the use of structuredmatrices in a compressed LSTM layer 120 of a recurrent neural network110 will be discussed using Toeplitz-like matrices as an example. Thecomplexity of Toeplitz-like matrices can be controlled using thedisplacement rank, i.e., the number of products that are being summedtogether as shown in sum of the circulant and skew-circulant matricesFIG. 2B. Low displacement rank corresponds to highly structured matricessuch as circulant and Toeplitz matrices and their inverses. Highdisplacement rank matrices can be used to model increasinglyunstructured matrices. in some examples, the displacement rank can beused to control the computational complexity, storage requirements, andmodeling capacity of for a compression scheme. In some examples, thedisplacement rank can be tuned based on application requirements.

In some implementations, the Toeplitz-like matrix structure is appliedto recurrent and inter-layer parameter matrices for all of the gates ina particular compressed LSTM layer 120. In some implementations, theToeplitz-like matrix structure is applied to lower ordered layer(s) in alayer stack (e.g., layers 1 and 2).

In a projection compression model for a recurrent neural network layer,a compressed LSTM layer 120 is produced by replacing uncompressedrecurrent parameter matrix, of size m×n, from a particular layer (e.g.,layer l) with a compressed recurrent parameter matrix, of size m×r, anda projection matrix r×n. In addition, a corresponding inter-layerparameter matrix, also of size m×n, from a next higher order LSTM layer(e.g., layer l+1) is replaced by a compressed inter-layer matrix, ofsize m×r, and the same projection matrix. Furthermore, the respectiveranks of the compressed recurrent and inter-layer matrices and theprojection matrix are less than the ranks of the corresponding recurrentand inter-layer parameter matrices. The low-rank projection matrix isshared across the two corresponding layers. The projection compressionmodel can be applied to one or more different gates. In someimplementations, the projection compression model is applied to higherordered layer(s) in a layer stack (e.g., layers 2-N).

In some implementations, a layer or set of layers can be compressed bythe use of structured matrices (or structured-like matrices) and a layeror set of layers can be compressed using projection matrices. Forexample, a lower ordered layer or set of layers (e.g., layers 1 and 2)of an recurrent neural network 110 can be compressed by replacing gateparameter matrices with Toeplitz-like matrices and a higher orderedlayer or set of layers (e.g., layers 2-N) can be compressed usingprojection matrices.

In some implementations, the above described compression techniques mayresult in at least a 75% reduction in parameters for an LSTM neuralnetwork. In some implementations, where the system is a speechrecognition model, the above described compression techniques may resultin at least a 75% compression of the LSTM neural network while retaininga word error rate that is within 0.3% of a word error rate (WER) of anuncompressed version of the system. In some examples, the abovedescribed compression techniques may result a compression of an LSTMneural network that ranges from 75% -83% with a corresponding WER from0.3% -2.3%.

In order to configure the recurrent neural network 110 to generateneural network outputs, the neural network system 100 trains therecurrent neural network 110 to determine trained values of theparameters of the recurrent neural network 110, including determiningtrained values of the parameters of the saturating LSTM layer 120.Training the recurrent neural network is described in more detail belowwith reference to FIG. 5.

FIG. 3 is a flow diagram of an example process 300 for processing acurrent layer input to generate a next layer output. For convenience,the process 300 will be described as being performed by a saturatingLSTM layer implemented by a system of one or more computers located inone or more locations. For example, a saturating LSTM layer in a neuralnetwork system, e.g., the compressed LSTM layer 120 of the neuralnetwork system 100 of FIG. 1, appropriately programmed in accordancewith this specification, can perform the process 300.

The LSTM layer applies a forget gate to a gate input vector to generatea forget gate output vector (step 302). Applying a gate to a gate inputvector will be described in more detail below with reference to FIG. 4.

In some implementations, the LSTM layer generates the gate input vectorby concatenating the current layer input and the current layer output.In some other implementations, the LSTM layer is a peephole LSTM layerthat generates the gate input vector by concatenating the current layerinput, the current layer output, and the current layer state.

The LSTM layer applies an input gate to the gate input vector togenerate an input gate output vector (step 304). Applying a gate to agate input vector will be described in more detail below with referenceto FIG. 4. In some implementations, the input gate includes a structuredparameter matrix, e.g., a Toeplitz-like structured parameter matrix. Insome implementations, the input gate includes a compressed recurrent orinter-layer matrix and a corresponding projection matrix.

In some implementations, instead of applying the input gate to generatethe input gate output vector, the system uses the forget gate outputvector as the input gate output vector. That is, in someimplementations, the input gate is the same as the forget gate.

The LSTM layer applies an output gate to the gate input vector togenerate an output gate output vector (step 306). Applying a gate to agate input vector will be described in more detail below with referenceto FIG. 4. In some implementations, the output gate includes astructured parameter matrix, e.g., a Toeplitz-like structured parametermatrix. In some implementations, the output gate includes a compressedrecurrent or inter-layer matrix and a corresponding projection matrix.

The LSTM layer generates an intermediate cell state update vector fromthe current layer input and the current layer output (step 308). Inparticular, the LSTM layer processes the current layer input and thecurrent layer output using a neural network layer having an activationfunction that is a squashing function to generate the intermediate cellstate update vector.

Generally, a squashing function is a function that maps received inputsto a range of −1 to 1, exclusive. For example, the squashing functionmay be the hyperbolic tangent function.

The LSTM layer combines the intermediate cell state update vector andthe input gate output vector to generate a final cell state updatevector (step 310). In particular, the LSTM layer computes a point-wisemultiplication between the intermediate cell state update vector and theinput gate output vector to generate the final cell state update vector.

The LSTM layer combines the current cell state and the forget gateoutput vector to generate an intermediate new cell state (step 312). Inparticular, the LSTM layer computes a point-wise multiplication betweenthe current cell state and the forget output vector to generate theintermediate new cell state. In some implementations, the forget gateincludes a structured parameter matrix, e.g., a Toeplitz-like structuredparameter matrix. In some implementations, the forget gate includes acompressed recurrent or inter-layer matrix and a correspondingprojection matrix.

The LSTM layer combines, e.g., sums, the intermediate new cell state andthe final cell state update vector to generate a final new cell state(step 314).

The LSTM layer generates a new layer output from the final new cellstate (step 316). To generate the new layer output, the LSTM layerapplies a squashing function to each component of the final new cellstate to generate an intermediate new layer output.

The LSTM layer then combines the output gate output vector and theintermediate new layer output to generate the new layer output. Inparticular, the LSTM layer performs a pointwise multiplication betweenthe output gate output vector and the intermediate new layer output togenerate the new layer output.

In addition to using the final new layer state in generating the newlayer output, the LSTM layer maintains the final new cell state alongwith the new layer output for use at a subsequent time step.

FIG. 4 is a flow diagram of an example process 400 for applying a gateto a gate input vector to generate a gate output vector. Forconvenience, the process 400 will be described as being performed by asaturating LSTM layer implemented by a system of one or more computerslocated in one or more locations. For example, a compressed LSTM layerin a neural network system, e.g., the LSTM layer 120 of the neuralnetwork system 100 of FIG. 1, appropriately programmed in accordancewith this specification, can perform the process 400.

The LSTM layer determines the gate input vector (step 402).

The LSTM layer generates a respective intermediate gate output vectorfrom the gate input vector in accordance with a set of parameters (step404). In some implementations, the LSTM layer performs a matrixmultiplication between a parameter matrix and the gate input vector andthen adds a bias vector to the output of the matrix multiplication togenerate the intermediate gate output vector, with each of the gateshaving different parameter matrices and bias vectors. That is, inimplementations where the LSTM layer has a distinct input gate, forgetgate, and output gate, each of these gates will have different parametermatrices and bias vectors from each other gate.

The LSTM layer applies a gating function to each component of therespective intermediate gate output vector to generate a final gateoutput vector (step 406).

Generally, a gating function is a function that maps received inputs toa range of 0 to 1, exclusive. For example, a gating function may be thesigmoid function.

For at least one of the gates of the LSTM layer, however, the parametermatrix referenced in step 404 is a compressed parameter matrix. Anuncompressed parameter matrix is replaced with a compressed parametermatrix. The LSTM layer then performs a matrix multiplication between thecompressed parameter matrix and the gate input vector. For example, thecompressed matrix can replace a parameter matrix in any of an inputgate, a forget gate, a cell state, or an output gate. In someimplementations, compressed parameter matrices are applied to multiplegates in an LSTM layer. For example, compressed parameter matrices canbe applied to both an input and an output gate. As another example,compressed parameter matrices can be applied to an input, an outputgate, and a forget gate.

In some implementations, the compressed parameter matrix is astructured-like matrix, e.g., a Toeplitz-like structured matrix. In someimplementations, an uncompressed gate parameter is re-defined with acompressed parameter matrix and a corresponding projection matrix.

FIG. 5 is a flow diagram of an example process 500 for training arecurrent neural network that includes a compressed LSTM layer. Forconvenience, the process 500 will be described as being performed by asystem of one or more computers located in one or more locations. Forexample, a neural network system, e.g., the neural network system 100 ofFIG. 1, appropriately programmed in accordance with this specification,can perform the process 500.

The system obtains training data for training the recurrent neuralnetwork (step 502). The training data includes multiple training examplepairs, with each training example pair including a training neuralnetwork input and a target neural network output for the training neuralnetwork input.

The system trains the recurrent neural network on the training data todetermine trained values of the parameters of the recurrent neuralnetwork from initial values of the parameters by optimizing, i.e.,either maximizing or minimizing, an objective function (step 504).During the training the system imposes constraints on the values of thecompressed matrix or matrices so that they continue to satisfy therequirements for the compressed matrix. For example, for Teoplitz-likestructured matrices the system can impose constraints so that theentries of each compressed matrix are always Teoplitz-like or, for theprojection model compressed matrices, the system can adjust the valuesof the projection matrix and the compressed parameter matrix rather thanadjusting the values of the parameter matrix directly.

The system trains the recurrent neural network by optimizing theobjective function using a conventional machine learning trainingtechnique, e.g., a stochastic gradient descent with backpropagationthrough time training technique. That is, the system can performmultiple iterations of the training technique to optimize the objectivefunction by adjusting the values of the parameters of the recurrentneural network.

Embodiments of the subject matter and the functional operationsdescribed in this specification can be implemented in digital electroniccircuitry, in tangibly-embodied computer software or firmware, incomputer hardware, including the structures disclosed in thisspecification and their structural equivalents, or in combinations ofone or more of them. Embodiments of the subject matter described in thisspecification can be implemented as one or more computer programs, i.e.,one or more modules of computer program instructions encoded on atangible non-transitory program carrier for execution by, or to controlthe operation of, data processing apparatus. Alternatively or inaddition, the program instructions can be encoded on anartificially-generated propagated signal, e.g., a machine-generatedelectrical, optical, or electromagnetic signal, that is generated toencode information for transmission to suitable receiver apparatus forexecution by a data processing apparatus. The computer storage mediumcan be a machine-readable storage device, a machine-readable storagesubstrate, a random or serial access memory device, or a combination ofone or more of them.

The term “data processing apparatus” refers to data processing hardwareand encompasses all kinds of apparatus, devices, and machines forprocessing data, including by way of example a programmable processor, acomputer, or multiple processors or computers. The apparatus can also beor further include special purpose logic circuitry, e.g., an FPGA (fieldprogrammable gate array) or an ASIC (application-specific integratedcircuit). The apparatus can optionally include, in addition to hardware,code that creates an execution environment for computer programs, e.g.,code that constitutes processor firmware, a protocol stack, a databasemanagement system, an operating system, or a combination of one or moreof them.

A computer program (which may also be referred to or described as aprogram, software, a software application, a module, a software module,a script, or code) can be written in any form of programming language,including compiled or interpreted languages, or declarative orprocedural languages, and it can be deployed in any form, including as astand-alone program or as a module, component, subroutine, or other unitsuitable for use in a computing environment. A computer program may, butneed not, correspond to a file in a file system. A program can be storedin a portion of a file that holds other programs or data, e.g., one ormore scripts stored in a markup language document, in a single filededicated to the program in question, or in multiple coordinated files,e.g., files that store one or more modules, sub-programs, or portions ofcode. A computer program can be deployed to be executed on one computeror on multiple computers that are located at one site or distributedacross multiple sites and interconnected by a communication network.

The processes and logic flows described in this specification can beperformed by one or more programmable computers executing one or morecomputer programs to perform functions by operating on input data andgenerating output. The processes and logic flows can also be performedby, and apparatus can also be implemented as, special purpose logiccircuitry, e.g., an FPGA (field programmable gate array) or an ASIC(application-specific integrated circuit).

Computers suitable for the execution of a computer program include, byway of example, can be based on general or special purposemicroprocessors or both, or any other kind of central processing unit.Generally, a central processing unit will receive instructions and datafrom a read-only memory or a random access memory or both. The essentialelements of a computer are a central processing unit for performing orexecuting instructions and one or more memory devices for storinginstructions and data. Generally, a computer will also include, or beoperatively coupled to receive data from or transfer data to, or both,one or more mass storage devices for storing data, e.g., magnetic,magneto-optical disks, or optical disks. However, a computer need nothave such devices. Moreover, a computer can be embedded in anotherdevice, e.g., a mobile telephone, a personal digital assistant (PDA), amobile audio or video player, a game console, a Global PositioningSystem (GPS) receiver, or a portable storage device, e.g., a universalserial bus (USB) flash drive, to name just a few.

Computer-readable media suitable for storing computer programinstructions and data include all forms of non-volatile memory, mediaand memory devices, including by way of example semiconductor memorydevices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks,e.g., internal hard disks or removable disks; magneto-optical disks; andCD-ROM and DVD-ROM disks. The processor and the memory can besupplemented by, or incorporated in, special purpose logic circuitry.

To provide for interaction with a user, embodiments of the subjectmatter described in this specification can be implemented on a computerhaving a display device, e.g., a CRT (cathode ray tube) or LCD (liquidcrystal display) monitor, for displaying information to the user and akeyboard and a pointing device, e.g., a mouse or a trackball, by whichthe user can provide input to the computer. Other kinds of devices canbe used to provide for interaction with a user as well; for example,feedback provided to the user can be any form of sensory feedback, e.g.,visual feedback, auditory feedback, or tactile feedback; and input fromthe user can be received in any form, including acoustic, speech, ortactile input. In addition, a computer can interact with a user bysending documents to and receiving documents from a device that is usedby the user; for example, by sending web pages to a web browser on auser's client device in response to requests received from the webbrowser.

Embodiments of the subject matter described in this specification can beimplemented in a computing system that includes a back-end component,e.g., as a data server, or that includes a middleware component, e.g.,an application server, or that includes a front-end component, e.g., aclient computer having a relationship graphical user interface or a Webbrowser through which a user can interact with an implementation of thesubject matter described in this specification, or any combination ofone or more such back-end, middleware, or front-end components. Thecomponents of the system can be interconnected by any form or medium ofdigital data communication, e.g., a communication network. Examples ofcommunication networks include a local area network (“LAN”) and a widearea network (“WAN”), e.g., the Internet.

The computing system can include clients and servers. A client andserver are generally remote from each other and typically interactthrough a communication network. The relationship of client and serverarises by virtue of computer programs running on the respectivecomputers and having a client-server relationship to each other.

While this specification contains many specific implementation details,these should not be construed as limitations on the scope of anyinvention or of what may be claimed, but rather as descriptions offeatures that may be specific to particular embodiments of particularinventions. Certain features that are described in this specification inthe context of separate embodiments can also be implemented incombination in a single embodiment. Conversely, various features thatare described in the context of a single embodiment can also beimplemented in multiple embodiments separately or in any suitablesubcombination. Moreover, although features may be described above asacting in certain combinations and even initially claimed as such, oneor more features from a claimed combination can in some cases be excisedfrom the combination, and the claimed combination may be directed to asubcombination or variation of a subcombination.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understood as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. In certain circumstances, multitasking and parallel processingmay be advantageous. Moreover, the separation of various system modulesand components in the embodiments described above should not beunderstood as requiring such separation in all embodiments, and itshould be understood that the described program components and systemscan generally be integrated together in a single software product orpackaged into multiple software products.

Particular embodiments of the subject matter have been described. Otherembodiments are within the scope of the following claims. For example,the actions recited in the claims can be performed in a different orderand still achieve desirable results. As one example, the processesdepicted in the accompanying figures do not necessarily require theparticular order shown, or sequential order, to achieve desirableresults. In certain implementations, multitasking and parallelprocessing may be advantageous.

What is claimed is:
 1. A system for receiving a system input comprisinga respective neural network input at each of a plurality of time stepsand providing, in response to the received system input, a system outputcomprising a respective neural network output at each of the pluralityof time steps, the system comprising: a recurrent neural networkimplemented by one or more computers, wherein the recurrent neuralnetwork is configured to receive the respective neural network input ateach of the plurality of time steps and to generate the respectiveneural network output at each of the plurality of time steps, andwherein the recurrent neural network comprises: a first long short-termmemory (LSTM) layer, wherein the first LSTM layer is configured to, foreach of the plurality of time steps, generate a new layer state and anew layer output by applying a plurality of gates to a current layerinput, a current layer state, and a current layer output, each of theplurality of gates being configured to, for each of the plurality oftime steps, generate a respective intermediate gate output vector bymultiplying a gate input vector and a gate parameter matrix, and whereinthe gate parameter matrix for at least one of the plurality of gates isa Toeplitz-like structured matrix.
 2. The system of claim 1, wherein therecurrent neural network comprises a second LSTM layer, wherein thesecond LSTM layer is configured to, for each of the plurality of timesteps, generate a second new layer state and a second new layer outputby applying a second plurality of gates to a second current layer input,a second current layer state, and a second current layer output, each ofthe second plurality of gates being configured to, for each of theplurality of time steps, generate a respective second intermediate gateoutput vector by multiplying a second gate input vector and a secondgate parameter matrix, and wherein the gate parameter matrix for atleast one of the second plurality of gates is defined by a compressedparameter matrix and a projection matrix.
 3. The system of claim 2,wherein the first LSTM layer and the second LSTM layer are each one of aplurality of LSTM layers in an ordered stack of layers.
 4. The system ofclaim 3, wherein the first LSTM layer is lower in the stack than thesecond LSTM layer.
 5. The system of claim 1, wherein each of theplurality of gates are configured to, for each of the plurality of timesteps, apply a respective gating function to each component of therespective intermediate gate output vector to generate a respectivefinal gate output vector.
 6. The system of claim 1, wherein the neuralnetwork is an acoustic model.
 7. The system of claim 1, wherein theneural network is a speech recognition model.
 8. The system of claim 1,wherein the neural network is compressed by at least 75% of anuncompressed version of the neural network.
 9. The system of claim 1,wherein a word error rate of the neural network is within 0.3% of a worderror rate of an uncompressed version of the neural network.
 10. Asystem for receiving a system input comprising a respective neuralnetwork input at each of a plurality of time steps and providing, inresponse to the received system input, a system output comprising arespective neural network output at each of the plurality of time steps,the system comprising: a recurrent neural network implemented by one ormore computers, wherein the recurrent neural network is configured toreceive the respective neural network input at each of the plurality oftime steps and to generate the respective neural network output at eachof the plurality of time steps, and wherein the recurrent neural networkcomprises: a first long short-term memory (LSTM) layer, wherein thefirst LSTM layer is configured to, for each of the plurality of timesteps, generate a new layer state and a new layer output by applying aplurality of gates to a current layer input, a current layer state, anda current layer output, each of the plurality of gates being configuredto, for each of the plurality of time steps, generate a respectiveintermediate gate output vector by multiplying a gate input vector and agate parameter matrix, and wherein the gate parameter matrix for atleast one of the plurality of gates is defined by a compressed parametermatrix and a projection matrix.
 11. The system of claim 10, wherein therecurrent neural network comprises a second LSTM layer, wherein thesecond LSTM layer is configured to, for each of the plurality of timesteps, generate a second new layer state and a second new layer outputby applying a second plurality of gates to a second current layer input,a second current layer state, and a second current layer output, each ofthe second plurality of gates being configured to, for each of theplurality of time steps, generate a respective second intermediate gateoutput vector by multiplying a second gate input vector and a secondgate parameter matrix, and wherein the gate parameter matrix for atleast one of the second plurality of gates is a Toeplitz-like structuredmatrix.
 12. The system of claim 11, wherein the first LSTM layer and thesecond LSTM layer are each one of a plurality of LSTM layers in anordered stack of layers.
 13. The system of claim 12, wherein the secondLSTM layer is lower in the stack than the first LSTM layer.
 14. Thesystem of claim 10, wherein each of the plurality of gates areconfigured to, for each of the plurality of time steps, apply arespective gating function to each component of the respectiveintermediate gate output vector to generate a respective final gateoutput vector.
 15. The system of claim 10, wherein the neural network isan acoustic model.
 16. The system of claim 10, wherein the neuralnetwork is a speech recognition model.
 17. The system of claim 10,wherein the neural network is compressed by at least 75% of anuncompressed version of the neural network.
 18. The system of claim 10,wherein a word error rate of the neural network is within 0.3% of a worderror rate of an uncompressed version of the neural network.
 19. One ormore non-transitory computer storage media encoded with a computerprogram product, the computer program product comprising instructionsthat when executed by one or more computers cause the one or morecomputers to implement a system for receiving a system input comprisinga respective neural network input at each of a plurality of time stepsand providing, in response to the received system input, a system outputcomprising a respective neural network output at each of the pluralityof time steps, the system comprising: a recurrent neural network,wherein the recurrent neural network is configured to receive therespective neural network input at each of the plurality of time stepsand to generate the respective neural network output at each of theplurality of time steps, and wherein the recurrent neural networkcomprises: a first long short-term memory (LSTM) layer, wherein thefirst LSTM layer is configured to, for each of the plurality of timesteps, generate a new layer state and a new layer output by applying aplurality of gates to a current layer input, a current layer state, anda current layer output, each of the plurality of gates being configuredto, for each of the plurality of time steps, generate a respectiveintermediate gate output vector by multiplying a gate input vector and agate parameter matrix, and wherein the gate parameter matrix for atleast one of the plurality of gates is a Toeplitz-like structuredmatrix.
 20. One or more non-transitory computer storage media encodedwith a computer program product, the computer program product comprisinginstructions that when executed by one or more computers cause the oneor more computers to implement a system for receiving a system inputcomprising a respective neural network input at each of a plurality oftime steps and providing, in response to the received system input, asystem output comprising a respective neural network output at each ofthe plurality of time steps, the system comprising: a recurrent neuralnetwork, wherein the recurrent neural network is configured to receivethe respective neural network input at each of the plurality of timesteps and to generate the respective neural network output at each ofthe plurality of time steps, and wherein the recurrent neural networkcomprises: a first long short-term memory (LSTM) layer, wherein thefirst LSTM layer is configured to, for each of the plurality of timesteps, generate a new layer state and a new layer output by applying aplurality of gates to a current layer input, a current layer state, anda current layer output, each of the plurality of gates being configuredto, for each of the plurality of time steps, generate a respectiveintermediate gate output vector by multiplying a gate input vector and agate parameter matrix, and wherein the gate parameter matrix for atleast one of the plurality of gates is defined by a compressed parametermatrix and a projection matrix.